Although there have been significant recent efforts to develop methodologies and theories for high dimensional data analysis, there is a paucity of methods with theoretical guarantees for statistical inference with missing data in the high-dimensional setting. Under the assumption that the components are missing uniformly and completely at random (MUCR), Loh and Wainwright [1]} proposed a non-convex optimization approach to high-dimensional linear regression, Lounici [2]} introduced a method for estimating a low-rank covariance matrix and Lounici [3]} considered sparse principal component analysis. In these papers, theoretical properties of the procedures were analyzed. These methods and theoretical results critically depend on the MUCR assumption.
